{
  "schema": "AN04-restored-compact-tube-proof-map/v1",
  "proofs": [
    {
      "id": "TUBE:Q0",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "## 1.",
      "dependencies": [
        "G2:G0",
        "G2:G41",
        "U001:P2",
        "U001:P3"
      ],
      "scope": "Exact cotangent signs, actual quotient atlas, radial kernel and nontrivial closed-interval hypotheses"
    },
    {
      "id": "TUBE:Q1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "**Proof.** The homogeneous Hamilton identity",
      "dependencies": [
        "TUBE:Q0",
        "G2:NF1",
        "G2:NF7",
        "FN:R0",
        "U001:P2"
      ],
      "scope": "Projected autonomous uniqueness, nonradialness including endpoints and n1 exclusion"
    },
    {
      "id": "TUBE:D0",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "**Construct the map and its common domain.**",
      "dependencies": [
        "TUBE:Q1",
        "FN:R1",
        "PC:H1",
        "PC:H2",
        "PC:H3",
        "PC:H4",
        "PC:H5"
      ],
      "scope": "Marked homogeneous initial chart, Hamilton transport and normalized transverse coordinates"
    },
    {
      "id": "TUBE:D1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "Here is why one transverse ball",
      "dependencies": [
        "TUBE:D0",
        "G2:NF0",
        "G2:NF1",
        "G2:NF5",
        "G2:NF6",
        "G2:NF7",
        "U001:F0-COMP"
      ],
      "scope": "Complete finite continuation in both time directions, one parameter neighborhood and all derivatives"
    },
    {
      "id": "TUBE:D2",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "The Hamilton flow commutes",
      "dependencies": [
        "TUBE:D1",
        "G2:G1",
        "G2:G41",
        "G2:NF7"
      ],
      "scope": "Dilation extension for every positive radius, exact central characteristic, p=tau and initial chart agreement"
    },
    {
      "id": "TUBE:S1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "**Preservation of the symplectic form.**",
      "dependencies": [
        "TUBE:D2",
        "G2:G1",
        "G2:F2"
      ],
      "scope": "Full time-slice symplectic identity under actual flow"
    },
    {
      "id": "TUBE:S2",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "The mixed time components",
      "dependencies": [
        "TUBE:S1",
        "TUBE:D2",
        "G2:G0",
        "U001:P2",
        "U001:P3"
      ],
      "scope": "Correct mixed-time sign, full two-form identity and invertible differential"
    },
    {
      "id": "TUBE:S3",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "Homogeneity identifies the radial fields",
      "dependencies": [
        "TUBE:S2",
        "G2:G41",
        "G2:F1"
      ],
      "scope": "Exact primitive equality by radial contraction without unknown exact term"
    },
    {
      "id": "TUBE:I1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "**Global injectivity after one shrinking.**",
      "dependencies": [
        "TUBE:Q0",
        "TUBE:S2",
        "TUBE:D1",
        "U001:P3",
        "U001:F0-COMP"
      ],
      "scope": "Projected local inverse and compact limiting-collision contradiction including both endpoints"
    },
    {
      "id": "TUBE:I2",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "Take \\(V\\) to be the conic domain",
      "dependencies": [
        "TUBE:I1",
        "TUBE:D2",
        "TUBE:S2",
        "U001:P3"
      ],
      "scope": "Radial-scale uniqueness, full global injectivity and smooth open inverse"
    },
    {
      "id": "TUBE:I3",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "Finally \\(V\\) is parametrized",
      "dependencies": [
        "TUBE:I2",
        "U001:F0-COMP",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Explicit based contraction CTA1, time convexity, arbitrary graph-neighborhood shrinking and endpoints"
    },
    {
      "id": "TUBE:M1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "**Proof of the symbol choice.**",
      "dependencies": [
        "TUBE:I3",
        "GS:Z5",
        "GS:Z6",
        "MT:I0",
        "MT:I2",
        "U001:F0-COMP"
      ],
      "scope": "Intrinsic fourth-root flat line, compact-grid homotopy cancellation, actual nonzero global smooth flat section"
    },
    {
      "id": "TUBE:M2",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "Choose the section on the normalized slice",
      "dependencies": [
        "TUBE:M1",
        "GS:Z10",
        "GE:D0",
        "U001:P14.2-powers"
      ],
      "scope": "Dilation-invariant section, exact symplectic half-density degree, all derivatives and inverse-symbol estimates"
    },
    {
      "id": "TUBE:M3",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "Choose a smooth degree-zero cutoff",
      "dependencies": [
        "TUBE:M2",
        "GS:Z11",
        "GS:Z12",
        "GE:D2",
        "U001:P14.3",
        "U001:F0-COMP"
      ],
      "scope": "Finite local phase realization and homogeneous partition, exact transition gluing and arbitrary wavefront confinement"
    },
    {
      "id": "TUBE:M4",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "The input and output base images",
      "dependencies": [
        "TUBE:M3",
        "TUBE:D2",
        "U001:F0-COMP",
        "U001:P14.3",
        "RC:C1",
        "P2:OP5"
      ],
      "scope": "Uniform radial comparability on compact normalized support, no axes, compact proper kernel cutoffs"
    },
    {
      "id": "TUBE:M5",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "**Both inverses on one smaller tube.**",
      "dependencies": [
        "TUBE:M4",
        "GE:G0",
        "GE:D9",
        "RC:C9",
        "K:K3",
        "CH:T2",
        "P2:OP5",
        "P2:OP6"
      ],
      "scope": "Both graph parametrices on one matched tube, ordered inverse comparison and full smoothing ideal"
    },
    {
      "id": "TUBE:E1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "**Proof.** Take \\(A_1,B_1\\)",
      "dependencies": [
        "TUBE:M5",
        "GE:D10",
        "TUBE:D2",
        "P2:OP6"
      ],
      "scope": "Full ordered Egorov, arbitrary real graph order and actual Psi0 remainder without homogeneous lower-term assumption"
    },
    {
      "id": "TUBE:E2",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "Choose a time interval containing",
      "dependencies": [
        "TUBE:E1",
        "TUBE:I3",
        "GE:D12",
        "GE:D13",
        "GE:D14",
        "GE:D15",
        "GE:D16",
        "GE:D17",
        "GE:D18"
      ],
      "scope": "One finite product cylinder, exact global symbol extension and complete ordered transport/residual construction"
    },
    {
      "id": "TUBE:E3",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "This application uses the complete transport",
      "dependencies": [
        "TUBE:E2",
        "GE:D13",
        "GE:D14",
        "GE:D16",
        "GE:D17",
        "GE:D18"
      ],
      "scope": "Uniform exponential/inverse bounds, all ordinary derivatives, variation of constants and supported asymptotic sum on same domain"
    },
    {
      "id": "TUBE:E4",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "Define \\(A=A_1A_2\\)",
      "dependencies": [
        "TUBE:E3",
        "TUBE:M5",
        "RC:C9",
        "GE:D19",
        "CH:T2",
        "P2:OP6"
      ],
      "scope": "Both inverse and conjugation identities plus exact CT19/CT20 paired intertwining with every defect"
    },
    {
      "id": "TUBE:E5",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "The theorem treats scalar principal direction",
      "dependencies": [
        "TUBE:E4",
        "GE:D13",
        "GE:D19",
        "RP:P1"
      ],
      "scope": "Precise compatible finite-matrix extension and all-real-order reduction with original order-one flow time"
    },
    {
      "id": "TUBE:X1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "**Exercise 6.1",
      "dependencies": [
        "TUBE:Q0",
        "TUBE:I1",
        "TUBE:I2",
        "TUBE:S3",
        "G2:G0",
        "U001:P16.1",
        "U001:P16.2"
      ],
      "scope": "Unchanged complete periodic-ray solution including equality endpoint and exact cotangent lift"
    },
    {
      "id": "TUBE:X2",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "**Exercise 6.2",
      "dependencies": [
        "TUBE:Q1",
        "TUBE:D2",
        "G2:G0",
        "U001:P15.1",
        "G2:NF1"
      ],
      "scope": "Unchanged complete radial exponential example and independent-vector obstruction"
    },
    {
      "id": "TUBE:X3",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "**Exercise 6.3",
      "dependencies": [
        "TUBE:D2",
        "TUBE:S3",
        "U001:P16.1",
        "U001:P16.2",
        "U001:P16.3",
        "U001:P2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Unchanged complete tangent tube, exact flow/primitive, inverse, radial norm bounds and missing endpoint"
    },
    {
      "id": "TUBE:F1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "fdaf9bd9ecf02f01ca8bd7c95c2908ae084b3d294981b8f91fd512c2b851b0fc",
      "proof_locator": "The figure uses the explicit chart",
      "dependencies": [
        "TUBE:X3",
        "U001:P16.1",
        "U001:P16.2",
        "U001:P16.3"
      ],
      "scope": "Exact T1 base/covector diagram, positive minimum, true normalized probe and transverse omissions"
    }
  ],
  "earlier_maps": [
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  "public_release_authorized": false
}